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It is no more wrong to say that ∀x∈0:x∈T than it is to say that "all Leprechauns are Irish".
There is no claim that ∃x∈0:x∈T ("there exists an Irish leprechaun").

The negation of my claim is that ¬∀x∈0:x∈T ("not all leprechauns are Irish") which would have to be proven by showing that ∃x∈0:x∈T ("there exists a non-Irish leprechaun").

The (minority) mathematical faction of Constructivism holds that proof by negation is invalid; the obvious falsity of the negation is insufficient to proove the proposition. There seem to be more constructivists in this thread than have ever gathered before in one place.

An argument Harold already posed is relevent to address this stance:

The empty set is a proper subset of every other set. If an argument references a typed set, 0 must inherit the type. If it were allowed to not be of the referenced type, it could not have been a subset of the typed set. Harold is correct is typing 0 as exclusively prime...remembering that is can also be exclusively type as non-prime in another argument on another day!

Cheers!

[Sorry to bump so late in the game. I was dead-set against Harold and Bob (almost angry with them for their obstinance!) for a full hour of parsing this thread before their arguments convinced me that my intuitive distaste for their proposition was wrong.]

As you have pointed out, their properties can be used in a lot of algorithms. But this is the case for other "type" of numbers having other properties used in other type of algorithms. So your question is no easy to answer... It is like asking why knives are useful to cut something.

~RaGE();

I think words like 'destiny' are a way of trying to find order where none exists. - Christian Graus
Do not feed the troll ! - Common proverb

Hey Rage,
That was a sharp, and precise reply. Thanks.
The link for "Euclid's lemma" was helpful, however it builds other theorems based on the fundamental fact that " Any non prime number can be expressed as product of prime numbers".

Lastly, what an explanation..

Rage wrote:

As you have pointed out, their properties can be used in a lot of algorithms. But this is the case for other "type" of numbers having other properties used in other type of algorithms. So your question is no easy to answer... It is like asking why knives are useful to cut something.

I just loved it, But sadly.. This is what I want someone to answer for me..

It is not a unique ability for prime numbers, it is just that these numbers can not be sub-divided any further. You can find the LCM and HCF using any numbers, but they will always be a combination of prime factorials, so using prime numbers is far easier.

Vijay Sringeri wrote:

How is it possible that, any number can be expressed as product of prime factors ?

Essentially because a prime number can not be divided and a non prime number can be.

Any number n that is not prime has at least two divisors that are not 1 and n. These divisors are either prime or non prime. If they are non prime then by definition they follow the same rule as n. These numbers are smaller then n, so repeating this rule will always result in only prime divisors.

The complete lack of any mention of Zero in this discussion has sucked out all meaning for me, and left me inside a total vacuum.

Since Zero multiplied, or divided (except of course Zero divided by Zero), by any number, natural perverted, or even fractional, will always be Zero: therefore Zero is the Prime of Primes, not to mention that Zero raised to any power remains Zero, not to mention that subtracting Zero from, or adding Zero to, any number leaves the number unchanged !

That any number divided by Zero is an infinity (whose ordinality, or Aleph, among other possible infinities: is ultimate ?) which cannot be conceptualized within linearly digital Turing/Von Neumann theoretical computational design, and must be expressed by some "place-holder" like "undefined," or "NaN," or will, on a practical level, in many circumstances crash a computer: is proof of its sacred power.

Zero is the unique singularity of the transition between positive and negative numbers, thus equivalent to the Omphalos, the stone of the navel of the geo-body of the cosmos, which for the ancient Greeks was located at the shrine of the oracle at Delphi.

I propose to you that the infinite set of all possible prime numbers is contained within the infinity created by Zero divided by Zero like a tiny foot in a huge shoe: lots of wiggle-room no matter what #1 does, or does not, do.

best, Bill

"Takuan Sōhō died in Edo (present-day Tokyo) in December of 1645. At the moment before his death, Takuan painted the Chinese character 'meng' ("dream"), laid down his brush and died."

Because the "secret" factors A*B = C, with A prime and B prime, are the holy grail to break RSA, the asymmetric keys algorithm used world while to enforce security, with SSL, HTTPS, VPNs...

Ruffly speaking, in RSA itself, C is "the public key" and its factorization, A and B "the private key".

It is straightforward to find the factors for small numbers, but it happens that it is very hard to find such factors for large numbers (indeed - you need to extensively search for them).

To date it has been possible to break up to a 768 bit RSA key (C is 768 bits long) by using a cluster of many hundred servers. Larger keys (1024, 2048 bits) are still considered secure (needed hundred years of cluster computing time for one single key) - and they will, until some strong improvement will be performed in number theory.

Mathematicians love to decompose things. The parts of a thing a usually easier to handle.

When they considered the integers and the law of addition, they saw nothing interesting. Any integer larger than 2 decomposes into smaller parts: 12 decomposes into 1+11, 2+10, 3+9, 4+8, 5+7 and 6+6. This is trivial. 1 cannot be decomposed, so 1 is the only "prime" as regards addition.

When they considered the integers and the law of multiplication, things got funnier: 12 decomposes into 2 x 6 and 3 x 4, 6 decomposes into 2 x 3 and 4 decomposes into 2 x 2. But 2 and 3 cannot be decomposed.

So the idea of primes came quite naturally by discovering thoses numbers that cannot be decomposed. And they found much less regularity, meaning cool things to study.

There is no real mystery in the decomposition: either a number cannot be decomposed, then it is called prime, or it can be decomposed, then it is called composite. You can apply the same reasoning to its parts, which are smaller. In the end, you always end up with prime factors, because decomposition is a finite process. (This is a case of infinite descent[^].)

More mysterious is the fact that the decomposition is unique: whatever the way you apply the decomposition, you always end up with the same factors with the same multiplicity. Example: 12 = 2 x 6 = 2 x 2 x 3 or 12 = 3 x 4 = 3 x 2 x 2. But you easily understand that if a prime factor p appears r times in one decomposition of an integer n and s times in another (let r < s), we have a problem, as if we divide n by p^s, in the first case we will get a nonzero remainder and in the other case a zero remainder !

What makes prime numbers so attractive is that mathematicians have failed so far to find simple structures in their distribution, leading to the famous Rieman's hypothesis[^], considered a major unsolved problem in modern mathematics.

What is your question? The definition of primes is that they cannot be divided (within the set of whole numbers) by any other whole number other than 1 or the number itself. Everything else you write is an immediate consequence of that definition.

Ok, a bit more in detail:

- Any non-prime can be expressed as product of prime factors.
With the correction, this is really an immediate consequence and occasionally used as an alternate definition of what is a prime (as in: any number not expressable in that way is a prime)

- Prime factors can be used to check whether a number divides "N" ( N - Integer )
Lets say that N=n1*n2*n3, where n1, n2, n3 are the prime factors of N. If a number M is divisible by N, then there is some integer K, so that K=M/N
or:K=M/(n1*n2*n3)
Now multiply both sides with (n2*n3), and you getK*n2*n3 = M/n1
As you can see there is an integer K1=K*n2*n3 for which K1=M/n1, and therefore n1 is a divisor of M. The same is true for n2 and n3 - both are divisors of M. Now that you know that if N is a divisor of N, then its prime factors are also divisors, you can turn that knowledge around: You test whether M is divisable by n1! If you find out that M is not divisible by n1, it means that it cannot be divisible by N either!

- Prime factors can be used to find LCM and HCF
Given two numbers N and M, you may express them as a product of prime factors N=n1*n2*...*nk and M=m1*m2*...*ml. (if N or M is a prime, there will only be one number n1 or m1 respectively. But that doesn't stop the following conclusion) HCF is the product of the factors you get from intersecting the two sets of prime factors. If you multiply that number by anything else, it will either no longer be a divisor of N, or of M.

LCM is pretty similar, but I'm running out of time at this point In any case, this property follows from the definition as well.

A prime number is one that can't be divided by any other number other than itself or 1. I learned 1 was a prime, but excluded from the rules defining additional prime numbers because if it wasn't, there would only be one prime number: 1.
The definition of a prime number makes it special.

Vijay Sringeri wrote:

How is it possible that, any number can be expressed as product of prime factors?

One of the prime factors is 1. 1*3 is 3. ALL prime numbers can be multiplied by 1, therefore they are a product of prime factors. The rest of the numbers are products of prime numbers. (That you can then multiply 10 billion times by 1 if you want.)
Actually, not any number is a product, any INTEGER number is a product of primes. You can never get 3.14157 as a product of primes. I take it back, ONLY positive numbers can be derived from prime products. 0 is not a prime, nor is -1. You can forget about immaginary numbers too. In the lexicon of numbering systems, primes are really limited.

Vijay Sringeri wrote:

Why this unique ability for prime numbers ?

It's defined this way. (These statements also implies all the restrictions I mentioned.)